Question

Suppose that \( (X, Y) \) has joint probability mass function \[ P(X = 0, Y = 0) = P(X = 1, Y = 1) = \theta, \quad P(X = 1, Y = 0) = P(X = 0, Y = 1) = \frac{1}{2} - \theta, \] where \( 0 \leq \theta \leq \frac{1}{2} \) is an unknown parameter. Consider testing \( H_0 : \theta = \frac{1}{4} \) against \( H_1 : \theta = \frac{1}{3} \), based on a random sample \[ \{ (X_1, Y_1), (X_2, Y_2), \dots, (X_n, Y_n) \} \] from the above probability mass function. Let \( M \) be the cardinality of the set \[ \{ i : X_i = Y_i, 1 \leq i \leq n \}, \] If \( m \) is the observed value of \( M \), then which one of the following statements is true?

• A. The likelihood ratio test rejects \( H_0 \) if \( m>c \) for some \( c \)
• B. The likelihood ratio test rejects \( H_0 \) if \( m<c \) for some \( c \)
• C. The likelihood ratio test rejects \( H_0 \) if \( c_1<m<c_2 \) for some \( c_1 \) and \( c_2 \)
• D. The likelihood ratio test rejects \( H_0 \) if \( m<c_1 \) or \( m>c_2 \) for some \( c_1 \) and \( c_2 \)

Hint:

- The likelihood ratio test compares the likelihoods under the null and alternative hypotheses.
- The test rejects \( H_0 \) when the likelihood ratio exceeds a critical threshold, typically depending on the observed data.

Answer 1

The Correct Option is A

1) Understanding the likelihood ratio test:
The likelihood ratio test is based on comparing the likelihoods under the null hypothesis \( H_0 \) and the alternative hypothesis \( H_1 \). In this problem, we are testing \( H_0: \theta = \frac{1}{4} \) against \( H_1: \theta = \frac{1}{3} \). The likelihood ratio test compares the observed likelihood ratio with a critical value to decide whether to reject the null hypothesis.
2) Formulating the likelihood ratio:
The likelihood function for the observed sample is the product of the individual likelihoods for each pair \( (X_i, Y_i) \). Given the probability mass function, we can write the likelihood under \( H_0 \) and \( H_1 \). The likelihood ratio is: \[ \Lambda(m) = \frac{L(\theta = \frac{1}{3})}{L(\theta = \frac{1}{4})} \] The likelihood function depends on the number of matching pairs \( m \) (i.e., when \( X_i = Y_i \)). Since the likelihood ratio test involves comparing likelihoods, the test will reject \( H_0 \) if the likelihood ratio exceeds a certain threshold. This corresponds to \( m \) being greater than some critical value \( c \). 3) Conclusion:
Thus, the likelihood ratio test rejects \( H_0 \) when \( m>c \) for some constant \( c \). This matches option (A).